CLASSICAL AND NONCLASSICAL LIE SYMMETRIES OF THE . - UNCW Randall Library

it easier to find and use these variables and symmetries. Manifolds are advantageousto our cause, as they have features that make objects described in them “coordinatefree.” We will begin by describing concepts in topology and using those to definemanifolds.Next, we must consider the possibility that a differential equation may have morethan one set of coordinates that reduce it. In fact, it is quite common to find theexistence of several changes of variables that will make our system solvable. This isuseful because one invariance condition may be easier to utilize than the next. So,the new dilemma is being able to find all of the invariance conditions. With this inmind that we appeal to group theory in Chapter 3.Group theory was created as a means to study the solutions, or roots, of algebraicequations. The idea that the solutions of polynomial equations could be generatedfrom a group, called a symmetry group, was developed by Lagrange and furtherstudied by Galois. Part of group structure is a binary operation that maps elementsof the group to other elements of the group. Symmetry groups are simple examples ofthis phenomenon, where the operation is composition of functions and each elementdescribes a set of mappings from one element to the next. Groups in this setting arecalled discrete groups. For our purposes, continuous groups are beneficial since weneed to continually map curves under a given parameter.So now we need to describe invariant functions of continuous groups of transformations. Sophus Lie extended existing symmetry techniques of solving systemsof algebraic equations to solving systems of differential equations. The extensioninvolved describing a new kind of group, now called a Lie group, that carries withinit the underlying properties of a manifold.We begin in Chapter 3 by describing Lie groups and further expanding the list ofproperties that our groups must have. We start by redefining the original notion ofa group to make the properties behave more like transformations, where we think of3

the solution curves as being transformed, or mapped, into each other. We call these“groups of transformations” and in shaping groups to have these characteristics, wenow have a means to describe the directions that solution curves need to be mapped.So, if we can find a group that is connected to our system of equations that housesall invariant mappings of that system, then we have an efficient system of extractingthem.Finally, a process of using group theory presents itself: if we can find the groupof transformations for a system of differential equations, we can almost immediatelypull the solutions of the system out of the transformation group by using generatorsof that group. We establish this at the end of Chapter 3 by using what we willcall infinitesimal generators, which relate Lie groups to Lie algebras, by a mappingthat we call the exponential map. This connection proves pivotal for establishing ameans to actually calculate the infinitesimal generators, which we show in Chapter4 lead to reductions and solutions.Having presented most of the classical theory of Lie methods, we introduce inChapter 4 the process by which we may make the theory useful. The step-by-stepprocess is called prolongation, which involves expanding the infinitesimal generatorsso they act on a bigger space, called a jet space. We demonstrate this method mysolving Burgers’ equation using Lie symmetries. At the end of Chapter 4, we introduce a slightly altered version of the step-by-step process of classical symmetries,and add a condition to this process and the solutions to get so-called nonclassicalsymmetries.In the next chapter, Chapter 5, we finally discuss the topic at the epicenter of thiswork: the K(m, n) dispersion equation. Emphasis is placed on using Lie symmetrymethods to find and categorize its solutions in both the classical and nonclassicalsetting. The K(m, n) equation is a generalized version of the well-known KdVequation, which produces compacton solutions [14]. Compacton solutions are also4

known as solitary waves, with compact support, so that the solutions vanish outsidea finite region. Among the different solutions we may find for K(m, n), we suspectwe will find traveling wave solutions.5

2 ELEMENTARY DIFFERENTIAL GEOMETRYThe subject of solving differential equations with Lie symmetry groups involvesthe intersection of two considerably different topics of mathematics. Concepts inboth differential geometry and Lie algebra provide vital preparation for and insightinto the foundation of this solution technique. Before describing the solutions ofthese equations, or even the differential equations themselves, it is important toestablish the background. This is the role that differential geometry will play as wedescribe its relevance by way of manifolds.2.1 TopologyA manifold provides an environment for the objects they contain that is essentially “coordinate-free.” This proves very useful for the objects that we wish todescribe and solve. Defining a manifold requires some discussion of basic ideas intopology.Definition 2.1.1. A topological space is a non-empty set E together with a familyI (Ui i I) of subsets of E satisfying the following axiomsE E E I, I,J finite, J I \Ui I,i JJ I [Ui I.i JThe elements of I are called open relative to I, or of only one topology is used,just simply open. The pair (E, I) is called a topological space.6

Definition 2.1.2. Let (E, I) be a topological space. It is a Hausdorff space ifand only if it satisfies the following additional axiom:For every pair of distinct points x1 , x2 E there are disjoint neighborhoodsUi (xi ), i 1, 2 :( x1 , x2 E, x1 6 x2 )( U1 (x1 ), U2 (x2 )) : U1 (x1 ) U2 (x2 ) [13].Definition 2.1.3. A topological space is connected if it cannot be written as thedisjoint union of two open sets.2.2 ManifoldsAs we progress through this section, we will define manifolds and how mappingsact within their inherent structure. We can think of manifolds as a space that islocally Euclidean. In other words, if we “cut” out a part of a manifold, it will takethe form of well-known objects in Euclidean geometry. In this way, we describemanifolds as being stitched together with Euclidean patches, called charts, and wewant there to be as much overlapping as possible. This overlapping, as we will see,creates a nice differentiable structure that we may describe mappings upon.Definition 2.2.1. An m-dimensional manifold is a set M , together with a countable collection of subsets Uα M , called coordinate charts, and one-to-one functionsχα : Uα Vα onto connected open subsets Vα Rm , called local coordinate maps,which satisfy the following properties [7]:(a) The coordinate charts cover M :[Uα M .α(b) On the overlap of any pair of coordinate charts Uα Uβ the composite mapχβ χ 1α : χα (Uα Uβ ) χβ (Uα Uβ )7

is a smooth (infinitely differentiable) function.(c) If x Uα , x̃ Uβ are distinct points of M , then there exist open subsetsW Vα , W̃ Vβ , with χα (x) W , χβ (x̃) W̃ , satisfyingχ 1α (W ) χβ (W̃ ) Ø.The interactions of the manifold properties are illustrated in Figure 1.UβUαχαχβχβ χ 1αFigure 1: Charts on a manifold MExample 2.2.1. The Euclidean space Rm is a manifold. It has one coordinatechart U Rm and the identity chart χ Rm Rm . In addition, any open subsetU Rm is a m-dimensional manifold.8

2.3 Differentiable Maps and RankQuite often, we are interested in mappings on the differentiable structure of amanifold. We want these mappings to be smooth, so first we restrict the mappingsto differentiable maps. We would also like to draw attention to the characteristicsof differentiable maps. For theorems that we will discuss in Chapter 4, we mustdescribe what it means for differentiable maps to be of maximal rank.Definition 2.3.1. A differentiable map is a linear map such that the operations ofvector addition and scalar multiplication are preserved.The main properties of differential maps are [13](1) A constant map is differentiable at any point of its domain of definition.(2) A linear map B : Rn Rm is differentiable at any point x Rn and B 0 (x) B.(3) Differentiation formulae: Let f , g : u Rm be differentiable maps at x U Rn , then f g, f · g and λf, λ R, are differentiable at x, and we haved(f g)(x) df (x) dg(x),d(f · g)(x) f (x) · dg(x) g(x) · df (x),d(λf )(x) λdf (x).For the rest of this thesis, it will be convention to denote a multiplication f · gas simply f g.Definition 2.3.2. A differentiable mapping f : U Rm of an open subset U ofRn into Rm is said to be continuously differentiable or, of class C 1 (writtenf C 1 (U, Rm )) if9

df : U L(Rn , Rm )is a continuous map, i.e. df C 0 (U, L(Rn , Rm )) [13].Proposition 2.3.1. Let f : U Rm be a differentiable map on an open subset Uof Rn . The matrix of the differential df (x) is given with respect to the canonicalbases of Rn and Rm , respectively, by [13] (Aji ) f 1(x) · · · x1.m f(x) · · · x1 f 1(x) xn.m f(x) xn j f (x) , 1 i n; 1 j m. xi This m n matrix is referred to as the Jacobian matrix of f at the point x U.Definition 2.3.3. The rank of the mapping f at the point x U is defined to bethe rank of the Jacobian matrix at x [13].Theorem 2.3.1. Let F : M N be of maximal rank at x0 M . Then there arelocal coordinates x (x1 , . . . , xm ) near x0 , and y (y 1 , . . . , y n ) near y0 F (x0 )such that these coordinates F has the simple form [7]y (x1 , . . . , xm , 0, . . . , 0),if n m,ory (x1 , . . . , xn ),if n m.Definition 2.3.4. The map f : U V is a C 1 -diffeomorphism if [13](1) f C 1 (U, Rn );(2) f is bijective;(3) f 1 C 1 (V, Rn ).10

In order to extend these properties to functions of several variables, we have tointroduce higher-order differentials.Definition 2.3.5. Let f : U Rm be a map which is assumed to be differentiablein U Rn . Hence the derivativedf f 1 : U L(Rn , Rm )exists and is also differentiable. The map f : U Rm is said to be differentiableof order k on an open subset U of Rn , ifdk f d(dk 1 f ) : U Rn Lk (Rn , Rm ) L(Rn , L(Rn , . . . , L(Rn , Rm )); d0 f fexists. If dk f is continuous, f is said to be of class C k [13]. We define f to be C if it is C k for all k 0.Definition 2.3.6. The map f : U V, where U and V are open subsets of Rn , isa C k -diffeomorphism 0 k , if [13](1) f C k (U, Rn );(2) f is bijective;(3) f 1 C k (V, Rn ).The mappings that hold particular interest for us are ones that are one-to-oneand onto. We are defining an environment where all actions are mappings, and themappings are smooth and continuous. In order for the maps to be differentiable,we must have differentiable structure in the manifolds. We note that the degree ofsmoothness of a manifold M is determined by the degree of differentiability of theoverlap functions χβ χ 1α [12]. We are interested in smooth manifolds, so we requirethat the overlap functions be C diffeomorphisms.11

From here forward, we will add the restriction that manifolds must be of classC , making it analytic and smooth as a result. We will also say that manifolds areof constant dimension and call them differentiable manifolds.Definition 2.3.7. Let F : M N be a smooth mapping from an m-dimensionalmanifold M to an n-dimensional manifold N . The rank of F at a point x M isthe rank of the n m Jacobian matrix ( F i / xj ) at x, where y F (x) is expressedin any convenient local coordinates near x. The mapping F is of maximal rankon a subset S M if for each x S the rank of F is as large as possible (i.e., theminimum of m and n) [7].2.4 SubmanifoldsWhen defining objects acting on a manifold, often we are only interested in acertain section, or subset, of that manifold. When examining a subset, we must besure that it carries with it the intrinsic properties of the greater manifold. We callthese subsets submanifolds.Definition 2.4.1. Let M be a smooth manifold. A submanifold of M is a subsetN M , together with a smooth, one-to-one map φ : Ñ N M satisfying themaximal rank condition everywhere, where the parameter space Ñ is some othermanifold and N φ(Ñ ) is the image of φ. In particular, the dimension of N is thesame as that of Ñ , and does not exceed the dimension of M [7].The map φ is often called an immersion, so that a submanifold that contains itis called an immersed submanifold, otherwise known as a regular submanifold.2.5 Vector FieldsWe are interested in tangent vectors to solution curves and work towards definingthe “infinitesimal transformation”, where the solution curves to our system are ap12

propriately mapped. Suppose C is a smooth curve on a manifold M , parametrized byΦ : I M, where I is a subinterval of R. In local coordinates x (x1 , . . . , xm ), Cis given by m smooth functions φ(ε) (φ1 (ε), . . . , φm (ε)) of the real variable ε.Definition 2.5.1. At each point x φ(ε) of C the curve has a tangent vector,namely the derivative φ̇(ε) dφ/dε (φ̇1 (ε), . . . , φ̇m (ε)). In order to distinguishbetween tangent vectors and local coordinate expressions for points on the manifold,we adopt the notationv x φ̇1 (ε) φ̇2 (ε) 2 · · · φ̇m ( ) m1 x x xfor each tangent vector to C at x φ(ε)Two curves C {φ(ε)} and C̃ {φ̃(θ)} passing through the same pointx φ(ε ) φ̃(θ )for some ε , θ , have the same tangent vector if and only if their derivatives agreeat the point:dφ̃ dφ (ε ) (θ )dεdθFigure 2: Tangent VectorDefinition 2.5.2. The collection of all tangent vectors to all possible curves passingthrough a given point x in M is called the tangent space to M at x, and is denotedby T M x .13

Definition 2.5.3. The collection of all tangent spaces corresponding to all pointsx in M is called the tangent bundle of M , denoted by[TM T M x .x MDefinition 2.5.4. A vector field v on M assigns a tangent vector v x T M xto each point x M , with v x varying smoothly from point to point. In localcoordinates (x1 , . . . , xm ), a vector field has the formv x ξ 1 (x) 2m ξ(x) ··· ξ(x), x1 x2 xmwhere each ξ i (x) is a smooth function of x.At this point, we would like to begin connecting vectors to an ability to mapcurves. To do this, we must define integral curves and flows. Flows, in the end, arethe keys to solidifying the differential geometry aspect of Lie symmetries.Definition 2.5.5. An integral curve of a vector field v starting at x0 is a smoothparametrized curve x φ(ε) whose tangent vector at any point coincides with thevalue of a given tangent vector v at the same point x0 φ(0):φ̇(ε) v φ(ε)for all ε.In local coordinates, x φ(ε) (φ1 (ε), . . . , φm (ε)) must be a solution to theautonomous system of ordinary differential equationsdxi ξ i (x),dεi 1, . . . , m,(1)where the ξ i (x) are the components of v at x. For ξ i (x) smooth, the standardexistence and uniqueness theorems for systems of ordinary differential equations [7]guarantee that there is a unique solution to (1) for each set of initial data14

φ(0) x0 .This in turn implies the existence of a unique maximal integral curve passingthrough a given point, where “maximal” means that it is not contained in anylonger integral curve.2.5.1 FlowsIf v is a vector field, the parametrized maximal integral curve passing through xin M is given by Ψ(ε, x) and is called the flow generated by v.Definition 2.5.6. The flow of a vector field has the basic properties:Ψ(0, x) x,(2)dΨ(ε, x) v Ψ(ε,x)d (3)for all ε where defined, andΨ(δ, Ψ(ε, x)) Ψ(δ ε, x),x M(4)for all δ, ε R such that both sides of the equation are defined.Here, Property (3) states that v is tangent to the curve Ψ(ε, x) for fixed x atε 0. The condition that ε 0 carries over into all other aspects of Lie symmetrymethods, as flows prove essential to development. Property (4) says that if two flowsare composed, the resulting flow is simply the addition of the parameters of the twocomposed flows. In other words, flows are additive.15

2.5.2 Lie Brackets and Lie AlgebrasDefinition 2.5.7. Given two n n matrices A and B, the bracket (or commutator)of A and B, denoted [A, B], is defined to be [3][A, B] AB BA.Definition 2.5.8. A finite-dimensional real or complex Lie algebra is a finitedimensional real or complex vector space g, together with a map [·, ·] from g ginto g, with the following properties [3]:1. [·, ·] is bilinear.2. [X, Y ] [Y, X] for all X, Y g.3. [X, [Y, Z]] [Y, [Z, X]] [Z, [X, Y ]] 0 for all X, Y, Z g.Lie algebras are related to Lie groups, as we will discuss in the next chapter.16

3 LIE GROUPSIn the previous chapter, we defined vector fields and how they act on functions.We also very briefly discussed simple characteristics of Lie algebras. In this chapter,we will expand on the basic concept of groups and describe Lie groups. From Liegroups evolves the necessity of Lie groups of transformations, which is the backbone of mapping solution curves into each other. We then borrow concepts fromgroup theory and differential geometry to define one-parameter groups of transformations. One-parameter transformation groups contain all the invariant solutionsof our system of equations.3.1 r-Parameter Lie GroupsOur objective is to define Lie groups of transformations. So, let us start with thedefinition of a group and show how transformations have group structure.Definition 3.1.1.Definition 3.1.2. A group is a set G equipped with a binary operation such that[9](i) the associative law holds: for every x, y, z G,x (y z) (x y) z;(ii) there is an element e G, called the identity, with e x x x e for allx G;(iii) every x G has an inverse: there is x 1 G with x x 1 e x 1 x.17

We would now like to structure a group that also has the properties of a manifold.This ensures that mappings are continuously differentiable.Definition 3.1.3. An r-param

So now we need to describe invariant functions of continuous groups of trans-formations. Sophus Lie extended existing symmetry techniques of solving systems . We begin in Chapter 3 by describing Lie groups and further expanding the list of properties that our groups must have. We start by rede ning the original notion of